arXiv research

Boundary of fiber convex domains is a cohomological sphere.

Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.

In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.

Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connec…

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the…

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

Spherical T-duality for iterated sphere bundles

The study classifies manifolds with specific rational cohomology properties.

Study spherical T-duality and Massey products in iterated sphere bundles.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

Cohomology fractals illustrate complex 3-manifold properties.

The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …

New rack and multiple group rack cohomology for surfaces in 3-sphere.

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebrai…

Injective construction proves bounded cohomology dimensions.

The paper proves non-triviality of certain classes in sphere bundle cohomology.

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin…

We prove for closed, odd-dimensional GKM$_3$ manifolds of non-negative sectional curvature that both the equivariant and the ordinary rational cohomology split off the cohomology of an odd-dimensional sphere.

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

The paper studies Sasakian geometry on sphere bundles, focusing on extremal metrics and cohomology.

Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.

We give a group cohomological description of the Čech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina-Mess about hyperbolic groups. In case of a relatively hyperbolic Poincaré duality group pair, we show the Bowditch boundary is a homology manifold. For a thr…

The loop space LP_1 of the Riemann sphere is an infinite dimensional complex manifold consisting of maps (loops) from S^1 to P_1 in some fixed C^k or Sobolev W^{k,p} space. In this paper we compute the Dolbeault cohomology groups H^{0,1}(LP_1).

The paper proves a cohomological injection for a specific group.

Constructs hyperbolic reflection groups with 3D limit sets.

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

Study cohomology of homeomorphisms and diffeomorphisms of manifolds.

We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain s…

We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.

New metric found for 4-manifolds with specific properties.

This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove th…

Unified theory of orbifolds and cohomology.

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions wi…

Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.

We prove the quantum filtration on the Khovanov-Rozansky link cohomology H_p with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to H_p that is invariant under Reidemeister moves, whose E_1 term is isomorphic to the Khovanov-Roz…

Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.

We study cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$ when $\partial M$ is diffeomorphic to a torus or a sphere. In particular, we show that for a $3$-manifold $M$ with torus boundary which is not diffeomorphic to a solid torus, the torus …

In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. We show that if N is the exterior of a link in a rational homology sphere, then the Thu…

New conditions for circular orderability of direct products, linking to left-orderability of groups.

If $G_1$ and $G_2$ are finite groups with periodic Tate cohomology, then $G_1\times G_2$ acts freely and smoothly on some product $S^n \times S^n$.

Classifies certain high-dimensional manifolds with specific cohomology properties.

New cohomology functors refine classical invariants of homotopy types.

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold coverin…

Generalizes Floer homotopy via Morse-Bott theory.

Develops obstruction theory for a specific 4-manifold index.